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Notes on Number Theory and Discrete Mathematics Vol. 19, 2013, No. 1, 66–72 The structure of the Fibonacci numbers in the modular ring Z5 J. V. Leyendekkers1 and A. G. Shannon2 1 Faculty of Science, The University of Sydney Sydney, NSW 2006, Australia 2 Faculty of Engineering & IT, University of Technology Sydney, NSW 2007, Australia e-mails: [email protected], [email protected] Abstract: Various Fibonacci number identities are analyzed in terms of their underlying integer structure in the modular ring Z5. Keywords: Fibonacci sequence, Golden Ratio, modular rings, Binet formula. AMS Classification: 11B39, 11B50. 1 Introduction Over eight centuries ago in Chapter XII of his book, Liber Abaci, Leonardo of Pisa (nicknamed Fibonacci) presented and solved his famous problem on the reproduction of rabbits in terms of the famous sequence which bears his name. Four centuries later, Albert Girard in 1634 gave the notation for the recurrence relation for the terms of the sequence in use today, namely Fn+1 = Fn + Fn−1 . (1.1) Over the centuries since the Fibonacci sequence of integers has been applied to a myriad of mathematical applications, especially in number theory [1]. In particular, Kepler [6] observed that the ratio of consecutive Fibonacci numbers converges to the Golden Ratio φ. He also showed that the square of any term differs by unity from the product of the two adjacent terms in the sequence (Simson’s or Cassini’s Identity (3.2) below). In this paper we analyse the structure of the Fibonacci sequence in the context of the modular ring Z5 (Table 1) [2]. The underlying structure accounts for many of the unique properties of this fascinating sequence, particularly their congruence properties [7]. 66 Class Row 0 1 2 3 4 05 5r0 0 5 10 15 20 15 5r1+1 1 6 11 16 21 25 5r2+2 2 7 12 17 22 35 5r3+3 3 8 13 18 23 45 5r4+4 4 9 14 19 24 Table 1. Rows of modular ring Z5 2 Class patterns of Fibonacci numbers The pattern of the Fibonacci numbers in Z5 is displayed in Table 2. n 1 2 3 4 5 6 7 8 9 Z5 15 15 25 35 05 35 35 15 45 10 11 12 13 14 15 16 17 18 19 05 45 45 35 25 05 25 25 45 15 n 20 21 22 23 24 25 26 27 28 Z5 05 15 15 25 35 05 35 35 15 29 30 31 32 33 34 35 36 37 38 45 05 45 45 35 25 05 25 25 45 Table 2. Fibonacci numbers in Z5 The patterns of the modular residues follow the form N 5 0 5 N 5 N 5 M 5 in which the numbers N 5 have the pattern 15 35 4 5 2 5 and the interstitial numbers M 5 have the pattern 2 515 35 4 5 . These patterns allow prediction of the class of Fn, and hence the right-end-digit (RED) from n (Table 3). Fn* (1,6) 15 (2,7) 25 (3,8) 35 (4,9) 45 (0,5) 05 n for N 5 19, 39, 59, 79, ... 1, 21, 41, 61, ... 2, 22, 42, 62, ... 14, 34, 54, 74, ... 16, 36, 56, 76, ... 17, 37, 57, 77, ... 4, 24, 44, 64, ... 6, 26, 46, 66, ... 7, 27, 47, 67, ... 9, 29, 49, 69, ... 11, 31, 51, 71, ... 12, 32, 52, 72, ... 0,5,10,15,20,... n for M 5 8, 28, 48, 68, ... 3, 23, 43, 63, ... 13, 33, 53, 73, ... 18, 38, 58, 78, ... Table 3. Details of the patterns (Fn*: Class of Fn) 67 There are many characteristics of the Fibonacci sequence that are directly related to this structure. We consider some of them here. They serve as examples for further analysis. Another approach would be to consider the algebra of F( 5 ) where F(x) is the characteristic polynomial associated with the recurrence relation (1.1) and is irreducible in the field F of its coefficients [8]. 3 The relationship to the Golden Ratio If the measure of a line AB is given by Fn+1,nd AB is divided into two different sized segments, AC and CB, with CB>AC, then AB/CB = CB/AC approximately defines φ, the Golden Ratio if CB = Fn and AC = Fn-1, so that approximately Fn+1 Fn−1 ≅ Fn2 (3.1) However, as first noted by Kepler the two sides of (3.1) always differ by unity as we can see from the class structures (Table 4). n Fn* (F ) Fn*+1 Fn*−1 (Fn+1 Fn−1 )* 4 41 77 92 3,8 1,6 2,7 4,9 9,4 1,6 4,9 6,1 0,5 1,6 4,9 3,8 2,7 0,5 2,7 4,9 0,5 0,5 3,8 7,2 2 * n Table 4. Data from Tables 2, 3 The equality is expressed in Simson’s Identity Fn+1 Fn−1 = Fn2 + (−1) n (3.2) Fn+1 / Fn = Fn / Fn−1 + (−1) n / (Fn Fn−1 ) (3.3) of which the second term on the right hand side is very small for large n; this is the error term in the Fibonacci approximation for φ [5]. Fn+6 = 4 Fn+3 + Fn (3.4) Fn+6 / Fn − 4 Fn+3 / Fn = 1, (3.5) so that which when substituted into (3.2) yield ⎧ Fn3 + Fn+6 − 4 Fn+3 , n even, Fn+1 Fn Fn−1 = ⎨ 3 ⎩ Fn − Fn+6 + 4 Fn+3, n odd . Many other elegant relationships can be formed in this way. 68 (3.6) 4 “Squaring” rectangles An odd number of golden rectangles with sides equal to successive Fibonacci numbers can appear to fit into squares as “demonstrated” in Figure 1. 8 A 5 E 5 C B 5 5 8 F 8 A D E 3 3 F C 5 5 D 3 5 B Figure 1. “Squaring” the Golden Rectangle This is not drawn to scale, but essentially a golden rectangle of sides F5 and F7 units, and hence of area 65 square units, is transformed into a square of side F6 units and hence of area 64 square units. Of course, while the eye might just be deceived, Simson is not! From Simson’s identity we get for odd n that Fn2−1 + Fn Fn−1 + Fn Fn+1 = Fn2+1 (4.1) The structure of the Fibonacci sequence in Z5 in Section 2 shows that this square sum depends on the constraints of the squares which occur only in Classes 0 5 , 15 and 4 5 , and the sums are also confined to these classes in harmony with this square (Table 5). Classes of Number of Products n Fn+1 Fn2+1 3 5 7 9 11 13 15 17 19 21 35 35 15 05 45 25 25 45 05 15 45 45 15 05 15 45 45 15 05 15 2 n −1 F Class of + Fn Fn −1 + Fn Fn +1 15 + 2 5 + 15 = 4 5 45 + 05 + 05 = 45 4 5 + 4 5 + 35 = 15 15 + 4 5 + 0 5 = 0 5 0 5 + 0 5 + 15 = 15 15 + 2 5 + 15 = 4 5 45 + 05 + 05 = 45 4 5 + 4 5 + 35 = 15 15 + 4 5 + 0 5 = 0 5 0 5 + 0 5 + 15 = 15 Table 5. Classes of Sums from (4.1) 69 5 The Factor 11 The result lim Fn+6 / Fn = 8ϕ + 5 n →∞ from [4] was obtained from the above characteristics of the sequence. Moreover, the Class of the sum of ten consecutive integers is the same as the class of the seventh number in the ten. The seventh number times 11 equals the sum of the ten. This is consistent with 11∈15 and 15 × a 5 = a 5 (Table 6). Note that the RED of the sum is the same as the RED of the seventh number, and since the RED of 11 is 1 it is the only integer to satisfy. Range of n Class of sum Class of 7th Integer, N7 Class of 15 × N 7 1 – 10 35 35 35 2 – 11 15 15 15 3 – 12 45 45 45 4 – 13 05 05 05 5 – 14 45 45 45 6 – 15 45 45 45 7 – 16 35 35 35 8 – 17 25 25 25 9 – 18 05 05 05 10 – 19 25 25 25 Table 6. Class structure in sets of 10 integers The class structure of the 7th number in each set of ten integers is: 35 15 45 05 45 45 35 25 05 25 which corresponds to F7 on the Fn class pattern in Section 2. 6 Periodicity of Fibonacci number right end digits A RED periodicity of 60 for integers was discovered in general in 1774 by Joseph Louis Lagrange [6]. However, this periodicity pattern is more complicated than previously assumed for the Fibonacci sequence. For even REDs the interval is 60, but for odd REDs the intervals can be 20 or 40 which indeed sum to 60, and for Class 0 5 the intervals are 30 (Table 7). 70 Class of Fn Fn* n* 05 0 0 30, 60, 90, 120, 150 30, 30, 30, 30 5 5 15, 45, 75, 105, 135 30, 30, 30, 30 6 1 21, 81, 141, 201 60, 60, 60, 60 2 42, 102, 162, 222 60, 60, 60, 60 8 48, 108, 168, 228 60, 60, 60, 60 9 39, 99, 159, 219 60, 60, 60, 60 1 1, 41, 61, 101, 121 40, 20, 40, 20 2 2, 22, 62, 82, 122 20, 40, 20, 40 8 8, 28, 68, 88, 128 20, 40, 20, 40 9 19, 59, 79, 119, 139 40, 20, 40, 20 3 3, 63, 123, 183 60, 60, 60, 60 4 54, 114, 174, 234 60, 60, 60, 60 6 36, 96, 156, 216 60, 60, 60, 60 7 57, 117, 177, 237 60, 60, 60, 60 3 23, 43, 83, 103, 143 20, 40, 20, 40 4 14, 34, 74, 94, 134 20, 40, 20, 40 6 16, 56, 76, 116, 136 40, 20, 40, 20 7 17, 37, 77, 97, 137 20, 40, 20, 40 3 33, 93, 153, 213 60, 60, 60, 60 4 24, 84, 144, 204 60, 60, 60, 60 6 6, 66, 126, 186 60, 60, 60, 60 7 27, 87.147, 207 60, 60, 60, 60 3 13, 53, 73, 113, 133 40, 20, 40, 20 4 4, 44, 64, 104, 124 40, 20, 40, 20 6 26, 46, 86, 106, 146 20, 40, 20, 40 7 7, 47, 67, 107, 127 40, 20, 40, 20 1 51, 111, 171, 231 60, 60, 60, 60 2 12, 72, 132, 192 60, 60, 60, 60 8 18, 78, 138, 198 60, 60, 60, 60 9 9, 69, 129, 189 60, 60, 60, 60 1 11, 31, 71, 91, 131 20, 40, 20, 40 2 32, 52, 92, 112, 152 20, 40, 20, 40 8 38, 58, 98, 118, 158 20, 40, 20, 40 9 29, 49, 89, 109, 149 20, 40, 20, 40 15 1 25 2 7 35 8 3 45 4 9 n Table 7. Periodicities of Fibonacci REDs 71 Δn 7 Final comments The structure of the Fn2 sequence is 1 5 15 4 5 4 5 0 5 4 5 4 5 15 1 5 0 5 15 15 4 5 4 5 0 5 4 5 4 5 which follows from the restricted distribution of the squares in Z5. This simple structure facilitates the formation of Pythagorean triples from Fn [1], and known results such as F2 n−1 = Fn2 + Fn2−1 , n > 1 , and n Fn+ 2 = 1 + ∑ F j . j =1 These can also be related to the structure. Finally, the interested reader might like to apply the foregoing to the Pellian sequences to compare the similarities and differences with the Fibonacci sequence [3]. References [1] Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No 9. [2] Leyendekkers, J.V., A.G. Shannon. The Modular Ring Z5. Notes on Number Theory and Discrete Mathematics. Vol. 18, 2012, No. 2, 28–33. [3] Leyendekkers, J.V., A.G. Shannon. Geometrical and Pellian Sequences. Advanced Studies in Contemporary Mathematics. Vol. 22, 2012, No. 4, 507–508. [4] Leyendekkers, J.V., A.G. Shannon. On the Golden Ratio (Submitted). [5] Leyendekkers, J.V., A.G. Shannon. The Decimal String of the Golden Ratio (Submitted). [6] Livio, Mario. The Golden Ratio. Golden Books, New York, 2002. [7] Shannon, A.G., A.F. Horadam, S.N. Collings. Some Fibonacci Congruences. The Fibonacci Quarterly. Vol. 12, 1974, No. 4, 351–354. [8] Simons, C.S., M. Wright. Fibonacci Imposters. International Journal of Mathematical Education in Science and Technology. Vol. 38, 2007, No. 5, 677–682. [9] Ward, M. The Algebra of Recurring Series. Annals of Mathematics. Vol. 32, 1931, No. 1, 1–9. 72